Q: Degree of the formulas for arithmetic sequence If we have $f(x)$ as a polynomial and $g(x) = f(1) +f(2) + \cdots + f(x)$ and x is any positive integer, I observed that the degree of $g(x)$ is always $1$ higher than the degree of $f(x)$. The only explanation I can think of is this:
If we think of $g(x)$ as $f(x-a) + f(x-b) + \cdots + f(x)$, there will be $x$ number of $x ^ n$ where $n$ is the highest degree of $f(x)$. Thus, summing it up, it simplifies to $x ^ {n + 1}$. I really do feel that something is off with this, can someone please help me?

A: Faulhaber's formula shows that $\sum_{k=1}^n k^p$ is a polynomial of $n$ of degree $p+1$. It follows that if $f(x) = \sum_{j=0}^p a_j x^j$ is a polynomial of degree $p$, then $\sum_{k=1}^n f(k)$ is a polynomial of $n$ of degree $p+1$.
